Parallel inexact Newton–Krylov and quasi-Newton solvers for nonlinear elasticity

A., Barnafi, Nicolás; Pavarino, Luca F.; Scacchi, Simone

doi:10.1016/j.cma.2022.115557

Cited by 11 publications

(2 citation statements)

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“…It is a common task in numerous disciplines (e.g., physics, chemistry, biology, economics, robotics, and engineering, social, and medical sciences) to construct a mathematical model with some parameters for an observed system which gives an observable response to an observable external effect. The unknown parameters of the mathematical model are determined so that the difference between the observed and the simulated system responses of the mathematical model for the same external effect is minimized (see e.g., [1][2][3][4][5][6][7][8][9][10][11]). This problem leads to finding the zero of a residual function (difference between observed and simulated responses).…”

Section: Introductionmentioning

confidence: 99%

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Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates

Berzi

2024

AppliedMath

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A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method φS=1.618… to super-quadratic φT=φS+1=2.618… and the quadratic convergence of the Newton method φN=2 to cubic φT=φN+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

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Section: Introductionmentioning

confidence: 99%

“…The suggested new strategy is based on Wolfe's [19] formulation of a generalized Secant method. The function x → f (x), where x ∈ R n and f : R n → R n , n > 1 (11) is locally replaced by linear interpolation through n + 1 interpolation base points A p , B p,k (k = 1, . .…”

Section: Introductionmentioning

confidence: 99%